CS145: INTRODUCTION TO DATA MINING Clustering Evaluation and Practical Issues Instructor: Yizhou Sun yzsun@cs.ucla.edu November 7, 2017
Learnt Clustering Methods Vector Data Set Data Sequence Data Text Data Logistic Regression; Naïve Bayes for Text Classification Decision Tree ; KNN; SVM ; NN K-means; hierarchical PLSA Clustering clustering; DBSCAN; Mixture Models Linear Regression Prediction GLM* Apriori; FP growth GSP; PrefixSpan Frequent Pattern Mining DTW Similarity Search 2
Evaluation and Other Practical Issues • Evaluation of Clustering • Model Selection • Summary 3
Measuring Clustering Quality • Two methods: extrinsic vs. intrinsic • Extrinsic: supervised, i.e., the ground truth is available • Compare a clustering against the ground truth using certain clustering quality measure • Ex. Purity, precision and recall metrics, normalized mutual information • Intrinsic: unsupervised, i.e., the ground truth is unavailable • Evaluate the goodness of a clustering by considering how well the clusters are separated, and how compact the clusters are • Ex. Silhouette coefficient 4
Purity • Let 𝑫 = 𝑑 1 ,… , 𝑑 𝐿 be the output clustering result, 𝜵 = 𝜕 1 ,… , 𝜕 𝐾 be the ground truth clustering result (ground truth class) • 𝑑 𝑙 𝑏𝑜𝑒 𝑥 𝑙 are sets of data points 1 • 𝑞𝑣𝑠𝑗𝑢𝑧 𝐷,Ω = 𝑂 σ 𝑙 max |𝑑 𝑙 ∩ 𝜕 𝑘 | 𝑘 • Match each output cluster 𝑑 𝑙 to the best ground truth cluster 𝜕 𝑘 • Examine the overlap of data points between the two matched clusters • Purity is the proportion of data points that are matched 5
Example • Clustering output: cluster 1, cluster 2, and cluster 3 • Ground truth clustering result: × ’s, ◊ ’s, and ○ ’s. • cluster 1 vs. × ’s, cluster 2 vs. ○ ’s, and cluster 3 vs. ◊ ’s 6
Normalized Mutual Information 𝐽(𝐷,Ω) • 𝑂𝑁𝐽 𝐷, Ω = 𝐼 𝐷 𝐼(Ω) 𝑄(𝑑 𝑙 ∩𝜕 𝑘 ) • 𝐽 Ω, 𝐷 = σ 𝑙 σ 𝑘 𝑄(𝑑 𝑙 ∩ 𝜕 𝑘 ) 𝑚𝑝 𝑄 𝑑 𝑙 𝑄(𝜕 𝑘 ) |𝑑 𝑙 ∩𝜕 𝑘 | 𝑂|𝑑 𝑙 ∩𝜕 𝑘 | = σ 𝑙 σ 𝑘 𝑚𝑝 𝑂 𝑑 𝑙 ⋅|𝜕 𝑘 | 𝐼 Ω = − σ 𝑘 𝑄 𝜕 𝑘 𝑚𝑝𝑄 𝜕 𝑘 • |𝜕 𝑘 | 𝑂 𝑚𝑝 |𝜕 𝑘 | = − 𝑂 𝑘 7
Example NMI=0.36 |𝝏 𝒍 ∩ 𝒅 𝒌 | |𝝏 𝒍 | Cluster 1 Cluster 2 Cluster 3 sum crosses 5 1 2 8 circles 1 4 0 5 diamonds 0 1 3 4 sum 6 6 5 N=17 |𝒅 𝒌 | 8
Precision and Recall • Random Index (RI) = (TP+TN)/(TP+FP+FN+TN) • F-measure: 2P*R/(P+R) • P = TP/(TP+FP) • R = TP/(TP+FN) • Consider pairs of data points: • hopefully, two data points that are in the same cluster will be clustered into the same cluster (TP), and two data points that are in different clusters will be clustered into different clusters (TN). Same cluster Different clusters Same class TP FN Different classes FP TN 9
Example Data points Output clustering Ground truth clustering (class) a 1 2 b 1 2 c 2 2 d 2 1 • # pairs of data points: 6 • (a, b): same class, same cluster • (a, c): same class, different cluster TP = 1 FP = 1 • (a, d): different class, different cluster FN = 2 • (b, c): same class, different cluster TN = 2 • (b, d): different class, different cluster RI = 0.5 • (c, d): different class, same cluster P= ½, R= 1/3, F = 0.4 10
Question • If we flip the ground truth cluster labels (2->1 and 1->2), will the evaluation results be the same? Data points Output clustering Ground truth clustering (class) a 1 2 b 1 2 c 2 2 d 2 1 11
Evaluation and Other Practical Issues • Evaluation of Clustering • Model Selection • Summary 12
Selecting K in K-means and GMM • Selecting K is a model selection problem • Methods • Heuristics-based methods • Penalty method • Cross-validation 13
Heuristic Approaches • For K-means, plot sum of squared error for different k • Bigger k always leads to smaller cost • Knee points suggest good candidates for k 14
Penalty Method: BIC • For model-based clustering, e.g., GMM, choose k that can maximizes BIC • Larger k increases the likelihood, but also increases the penalty term: a trade-off! 15
Cross-Validation Likelihood • The likelihood of the training data will increase when increasing k • Compute the likelihood on unseen data • For each possible k • Partition the data into training and test • Learn the GMM related parameters on training dataset and compute the log-likelihood on test dataset • Repeat this multiple times to get an average value • Select k that maximizes the average log-likelihood on test dataset 16
Evaluation and Other Practical Issues • Evaluation of Clustering • Model Selection • Summary 17
Summary • Evaluation of Clustering • Purity, NMI, F-measure • Model selection • How to select k for k-means and GMM 18
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